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Dynamic characteristics of a shrouded blade with impact and friction

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Abstract

A simplified computational model of a twisted shrouded blade with impact and friction is established. In this model, the shrouded blade is simulated by a flexible Timoshenko beam with a tip-mass, and the effects of centrifugal stiffening, spin softening, and Coriolis force are considered. Impact force is simulated using a linear spring model, and friction force is generated by a tangential spring model under sticking state and a Coulomb friction model under sliding state. The proposed model is validated by a finite element model. Then, the effects of initial gap and normal preload, coefficient of friction, and contact stiffness ratio (the ratio of tangential contact stiffness to normal contact stiffness) on system vibration responses are analyzed. Results show that resonant peaks become inconspicuous and impact plays a dominant role when initial gaps are large between adjacent shrouds. By contrast, in small initial gaps or initial normal preloads condition, resonant speed increases sharply, and the optimal initial normal preloads that can minimize resonant amplitude becomes apparent. Coefficient of friction affects the optimal initial normal preload, but it does not affect vibration responses when the contact between shrouds is under full stick. System resonant amplitude decreases with the increase of contact stiffness ratio, but the optimal initial normal preload is unaffected.

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Abbreviations

A :

Cross-sectional area of the blade

b :

Blade width

D, D*:

Rayleigh damping matrices before and after dimension reduction

E :

Young’s modulus

F, F̄ :

Canonical external force vectors without and with impact and friction

F*:

Canonical external force vector after dimension reduction

Fe, te]F0 :

Uniformly distributed aerodynamic force per unit length and aerodynamic force amplitude

Fy, Fz :

Components of impact and friction force in the flexural and swing directions

Ff1, Ff2 :

Friction force

fc(x):

Centrifugal force of the shrouded blade

f e :

Aerodynamic frequency

f r :

Rotational frequency

fn1, fn2 :

The first two-order natural frequencies of the shrouded blade

G, G*:

Coriolis force matrices before and after dimension reduction

h :

Blade thickness

Iy, Iz :

Area moment of inertias of y and z axes of the blade section

K, K*:

Stiffness matrices before and after stiffness reduction

Ke, Kc, Ks :

Structural stiffness matrix, centrifugal stiffening matrix, and spin softening matrix

K acc :

Dtiffness matrix caused by angular acceleration

kt, te]kn :

Tangential and normal contact stiffness

L :

Blade length

M, M*:

Mass matrices before and after dimension reduction

N :

Number of modal truncation

nr :

Dimension of matrix after dimension reduction

n :

Section number of the first FE model

N 0 :

Initial normal preload

N1, te]N2 :

Impact force

N*:

Dimension reduction matrix

m s :

Shroud mass

q, q*:

Canonical coordinate vectors of the blade before and after dimension reduction

R d :

Radius of disk

vL, wL :

Bending and swing displacements at blade tip

vs, ws :

Tangential and normal displacements of shroud

z s1 :

Tangential displacement of active blade shroud

zs2, zs3 :

Displacements of contact point

ρ :

Density of the blade

θ :

Rotational angle of the disk

β′:

Twist angle of shrouded blade

β1, βL :

Stagger angle at the root and blade tip of the blade

β n :

Angle of an arbitrary cross section between z axis and zn axis

β(x):

Twist angle of an arbitrary Point P of the blade

Ω:

Rotational speed/(r·min−1)

ϕ1i(x), ϕ2i(x), ϕ3i(x):

ith modal shape functions in radial, flexural/swing, and rotational angle directions

Δ:

Initial gap

μ :

Coefficient of friction

α :

Shroud inclination angle

ξ :

Contact stiffness ratio

ξ1, ξ2 :

First and second modal damping ratios

ω :

Angular velocity of the blade/(rad·s−1)

FE:

Finite element

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Acknowledgements

This project was supported by the National Natural Science Foundation (Grant No. 11772089), the Fundamental Research Funds for the Central Universities (Grant Nos. N170306004, N170308028, Nl 80708009, and Nl 80306005), the Program for the Innovative Talents of Higher Learning Institutions of Liaoning (Grant No. LR2017035), and Liaoning Revitalization Talents Program (Grant No. XLYC1807008).

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Xumin Guo, Jin Zeng, Hui Ma, Chenguang Zhao, Lin Qu & Bangchun Wen

  2. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, 110819, China

    Hui Ma

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  1. Xumin Guo
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Correspondence to Hui Ma.

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Guo, X., Zeng, J., Ma, H. et al. Dynamic characteristics of a shrouded blade with impact and friction. Front. Mech. Eng. 15, 209–226 (2020). https://doi.org/10.1007/s11465-019-0566-6

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